×

A tale of two Liouville closures. (English) Zbl 1387.12004

Summary: An \(H\)-field is a type of ordered valued differential field with a natural interaction between ordering, valuation, and derivation. The main examples are Hardy fields and fields of transseries. M. Aschenbrenner and L. van den Dries [Math. Z. 242, No. 3, 543–588 (2002; Zbl 1066.12002)] proved that every \(H\)-field \(K\) has either exactly one or exactly two Liouville closures up to isomorphism over \(K\), but the precise dividing line between these two cases was unknown. We prove here that this dividing line is determined by \(\lambda\)-freeness, a property of \(H\)-fields that prevents certain deviant behavior. In particular, we show that under certain types of extensions related to adjoining integrals and exponential integrals, the property of \(\lambda\)-freeness is preserved. In the proofs we introduce a new technique for studying \(H\)-fields, the yardstick argument which involves the rate of growth of pseudoconvergence.

MSC:

12H05 Differential algebra
12J10 Valued fields
06F20 Ordered abelian groups, Riesz groups, ordered linear spaces
12J15 Ordered fields
26A12 Rate of growth of functions, orders of infinity, slowly varying functions

Citations:

Zbl 1066.12002
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] ; Aschenbrenner, Asymptotic differential algebra and model theory of transseries. Asymptotic differential algebra and model theory of transseries. Annals of Mathematics Studies (2017) · Zbl 1430.12002
[2] ; Aschenbrenner, Valuation theory and its applications, II. Valuation theory and its applications, II. Fields Inst. Commun., 33, 7 (2003)
[3] 10.1006/jabr.1999.8128 · Zbl 0974.12015 · doi:10.1006/jabr.1999.8128
[4] 10.1007/s002090000358 · doi:10.1007/s002090000358
[5] 10.1090/conm/373/06914 · doi:10.1090/conm/373/06914
[6] ; Bourbaki, Fonctions d’une variable réelle, Chapitres IV-VII. Fonctions d’une variable réelle, Chapitres IV-VII. Actualités Sci. Ind., 1132 (1951)
[7] 10.1016/j.jalgebra.2016.08.016 · Zbl 1423.03133 · doi:10.1016/j.jalgebra.2016.08.016
[8] 10.1017/jsl.2016.59 · Zbl 1419.03035 · doi:10.1017/jsl.2016.59
[9] 10.1007/3-540-35590-1 · Zbl 1128.12008 · doi:10.1007/3-540-35590-1
[10] 10.1090/conm/171/01775 · doi:10.1090/conm/171/01775
[11] 10.1007/978-94-011-0443-2_27 · doi:10.1007/978-94-011-0443-2_27
[12] ; Kuhlmann, Ordered exponential fields. Ordered exponential fields. Fields Institute Monographs, 12 (2000) · Zbl 0989.12003
[13] 10.1007/BF01446522 · JFM 14.0369.04 · doi:10.1007/BF01446522
[14] 10.2307/2373949 · Zbl 0411.12021 · doi:10.2307/2373949
[15] 10.2140/pjm.1980.86.301 · Zbl 0401.12024 · doi:10.2140/pjm.1980.86.301
[16] 10.2307/2374255 · Zbl 0474.12020 · doi:10.2307/2374255
[17] 10.1016/0022-247X(83)90175-0 · Zbl 0518.12014 · doi:10.1016/0022-247X(83)90175-0
[18] 10.2307/1999639 · Zbl 0536.12015 · doi:10.2307/1999639
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.